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A solution to Ringel’s circle problem

  • James Davies

    University of Waterloo, Waterloo, Canada; University of Cambridge, Cambridge, UK

  • Chaya Keller

    Ariel University, Ariel, Israel

  • Linda Kleist

    Technische Universität Braunschweig, Braunschweig, Germany; Universität Potsdam, Potsdam, Germany

  • Shakhar Smorodinsky

    Ben-Gurion University of the Negev, Be’er-Sheva, Israel

  • Bartosz Walczak

    Jagiellonian University, Kraków, Poland
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Abstract

We construct families of circles in the plane such that their tangency graphs have arbitrarily large girth and chromatic number. This provides a strong negative answer to Ringel’s circle problem (1959). The proof relies on a (multidimensional) version of Gallai’s theorem with polynomial constraints, which we derive from the Hales–Jewett theorem and which may be of independent interest.

Cite this article

James Davies, Chaya Keller, Linda Kleist, Shakhar Smorodinsky, Bartosz Walczak, A solution to Ringel’s circle problem. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1557